?
G = C14.432+ (1+4) order 448 = 26·7
43rd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.432+ (1+4), C4⋊D4⋊17D7, C28⋊2D4⋊23C2, C4⋊C4.182D14, (D4×Dic7)⋊22C2, (C2×D4).156D14, C22⋊C4.49D14, C4.Dic14⋊19C2, Dic7⋊4D4⋊11C2, D14.30(C4○D4), C28.203(C4○D4), C4.96(D4⋊2D7), (C2×C14).158C24, (C2×C28).596C23, (C22×C4).225D14, C2.45(D4⋊6D14), C23.18(C22×D7), D14⋊C4.127C22, (D4×C14).124C22, C23.D14⋊19C2, C4⋊Dic7.372C22, (C22×C14).25C23, (C2×Dic7).77C23, (C4×Dic7).96C22, C22.179(C23×D7), C23.D7.26C22, C23.18D14⋊11C2, C23.21D14⋊26C2, Dic7⋊C4.139C22, (C22×C28).243C22, C7⋊7(C22.47C24), (C22×D7).191C23, (C22×Dic7).111C22, (D7×C4⋊C4)⋊23C2, (C4×C7⋊D4)⋊19C2, C2.42(D7×C4○D4), (C7×C4⋊D4)⋊20C2, (C2×C4×D7).86C22, C14.155(C2×C4○D4), C2.38(C2×D4⋊2D7), (C2×C4).40(C22×D7), (C7×C4⋊C4).146C22, (C2×C7⋊D4).31C22, (C7×C22⋊C4).15C22, SmallGroup(448,1067)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1004 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×10], C23, C23 [×2], C23, D7 [×2], C14 [×3], C14 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], Dic7 [×7], C28 [×2], C28 [×3], D14 [×2], D14 [×2], C2×C14, C2×C14 [×9], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D7 [×4], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×2], C7⋊D4 [×6], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7, C22×C14, C22×C14 [×2], C22.47C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4, Dic7⋊C4 [×4], C4⋊Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4, C23.D7, C23.D7 [×6], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7, C2×C4×D7 [×2], C22×Dic7 [×2], C2×C7⋊D4, C2×C7⋊D4 [×2], C22×C28, D4×C14, D4×C14 [×2], C23.D14 [×2], Dic7⋊4D4 [×2], C4.Dic14, D7×C4⋊C4, C23.21D14, C4×C7⋊D4, D4×Dic7, C23.18D14 [×2], C28⋊2D4, C28⋊2D4 [×2], C7×C4⋊D4, C14.432+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.47C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D7×C4○D4, C14.432+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=a7, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a7b-1, bd=db, ebe-1=a7b, cd=dc, ece-1=a7c, ede-1=b2d >
►On 224 points
Generators in S224
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 23 113 179)(2 22 114 178)(3 21 115 177)(4 20 116 176)(5 19 117 175)(6 18 118 174)(7 17 119 173)(8 16 120 172)(9 15 121 171)(10 28 122 170)(11 27 123 169)(12 26 124 182)(13 25 125 181)(14 24 126 180)(29 106 148 201)(30 105 149 200)(31 104 150 199)(32 103 151 198)(33 102 152 197)(34 101 153 210)(35 100 154 209)(36 99 141 208)(37 112 142 207)(38 111 143 206)(39 110 144 205)(40 109 145 204)(41 108 146 203)(42 107 147 202)(43 183 58 167)(44 196 59 166)(45 195 60 165)(46 194 61 164)(47 193 62 163)(48 192 63 162)(49 191 64 161)(50 190 65 160)(51 189 66 159)(52 188 67 158)(53 187 68 157)(54 186 69 156)(55 185 70 155)(56 184 57 168)(71 214 134 93)(72 213 135 92)(73 212 136 91)(74 211 137 90)(75 224 138 89)(76 223 139 88)(77 222 140 87)(78 221 127 86)(79 220 128 85)(80 219 129 98)(81 218 130 97)(82 217 131 96)(83 216 132 95)(84 215 133 94)
(1 192)(2 193)(3 194)(4 195)(5 196)(6 183)(7 184)(8 185)(9 186)(10 187)(11 188)(12 189)(13 190)(14 191)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 43)(26 44)(27 45)(28 46)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 71)(42 72)(57 180)(58 181)(59 182)(60 169)(61 170)(62 171)(63 172)(64 173)(65 174)(66 175)(67 176)(68 177)(69 178)(70 179)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 99)(92 100)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 153)(128 154)(129 141)(130 142)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)(139 151)(140 152)(197 215)(198 216)(199 217)(200 218)(201 219)(202 220)(203 221)(204 222)(205 223)(206 224)(207 211)(208 212)(209 213)(210 214)
(1 199 120 111)(2 200 121 112)(3 201 122 99)(4 202 123 100)(5 203 124 101)(6 204 125 102)(7 205 126 103)(8 206 113 104)(9 207 114 105)(10 208 115 106)(11 209 116 107)(12 210 117 108)(13 197 118 109)(14 198 119 110)(15 37 178 149)(16 38 179 150)(17 39 180 151)(18 40 181 152)(19 41 182 153)(20 42 169 154)(21 29 170 141)(22 30 171 142)(23 31 172 143)(24 32 173 144)(25 33 174 145)(26 34 175 146)(27 35 176 147)(28 36 177 148)(43 77 65 133)(44 78 66 134)(45 79 67 135)(46 80 68 136)(47 81 69 137)(48 82 70 138)(49 83 57 139)(50 84 58 140)(51 71 59 127)(52 72 60 128)(53 73 61 129)(54 74 62 130)(55 75 63 131)(56 76 64 132)(85 188 213 165)(86 189 214 166)(87 190 215 167)(88 191 216 168)(89 192 217 155)(90 193 218 156)(91 194 219 157)(92 195 220 158)(93 196 221 159)(94 183 222 160)(95 184 223 161)(96 185 224 162)(97 186 211 163)(98 187 212 164)
(1 63 8 70)(2 64 9 57)(3 65 10 58)(4 66 11 59)(5 67 12 60)(6 68 13 61)(7 69 14 62)(15 161 22 168)(16 162 23 155)(17 163 24 156)(18 164 25 157)(19 165 26 158)(20 166 27 159)(21 167 28 160)(29 222 36 215)(30 223 37 216)(31 224 38 217)(32 211 39 218)(33 212 40 219)(34 213 41 220)(35 214 42 221)(43 115 50 122)(44 116 51 123)(45 117 52 124)(46 118 53 125)(47 119 54 126)(48 120 55 113)(49 121 56 114)(71 209 78 202)(72 210 79 203)(73 197 80 204)(74 198 81 205)(75 199 82 206)(76 200 83 207)(77 201 84 208)(85 153 92 146)(86 154 93 147)(87 141 94 148)(88 142 95 149)(89 143 96 150)(90 144 97 151)(91 145 98 152)(99 140 106 133)(100 127 107 134)(101 128 108 135)(102 129 109 136)(103 130 110 137)(104 131 111 138)(105 132 112 139)(169 189 176 196)(170 190 177 183)(171 191 178 184)(172 192 179 185)(173 193 180 186)(174 194 181 187)(175 195 182 188)
G:=sub<Sym(224)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,23,113,179)(2,22,114,178)(3,21,115,177)(4,20,116,176)(5,19,117,175)(6,18,118,174)(7,17,119,173)(8,16,120,172)(9,15,121,171)(10,28,122,170)(11,27,123,169)(12,26,124,182)(13,25,125,181)(14,24,126,180)(29,106,148,201)(30,105,149,200)(31,104,150,199)(32,103,151,198)(33,102,152,197)(34,101,153,210)(35,100,154,209)(36,99,141,208)(37,112,142,207)(38,111,143,206)(39,110,144,205)(40,109,145,204)(41,108,146,203)(42,107,147,202)(43,183,58,167)(44,196,59,166)(45,195,60,165)(46,194,61,164)(47,193,62,163)(48,192,63,162)(49,191,64,161)(50,190,65,160)(51,189,66,159)(52,188,67,158)(53,187,68,157)(54,186,69,156)(55,185,70,155)(56,184,57,168)(71,214,134,93)(72,213,135,92)(73,212,136,91)(74,211,137,90)(75,224,138,89)(76,223,139,88)(77,222,140,87)(78,221,127,86)(79,220,128,85)(80,219,129,98)(81,218,130,97)(82,217,131,96)(83,216,132,95)(84,215,133,94), (1,192)(2,193)(3,194)(4,195)(5,196)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,43)(26,44)(27,45)(28,46)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,71)(42,72)(57,180)(58,181)(59,182)(60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,176)(68,177)(69,178)(70,179)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,153)(128,154)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,151)(140,152)(197,215)(198,216)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222)(205,223)(206,224)(207,211)(208,212)(209,213)(210,214), (1,199,120,111)(2,200,121,112)(3,201,122,99)(4,202,123,100)(5,203,124,101)(6,204,125,102)(7,205,126,103)(8,206,113,104)(9,207,114,105)(10,208,115,106)(11,209,116,107)(12,210,117,108)(13,197,118,109)(14,198,119,110)(15,37,178,149)(16,38,179,150)(17,39,180,151)(18,40,181,152)(19,41,182,153)(20,42,169,154)(21,29,170,141)(22,30,171,142)(23,31,172,143)(24,32,173,144)(25,33,174,145)(26,34,175,146)(27,35,176,147)(28,36,177,148)(43,77,65,133)(44,78,66,134)(45,79,67,135)(46,80,68,136)(47,81,69,137)(48,82,70,138)(49,83,57,139)(50,84,58,140)(51,71,59,127)(52,72,60,128)(53,73,61,129)(54,74,62,130)(55,75,63,131)(56,76,64,132)(85,188,213,165)(86,189,214,166)(87,190,215,167)(88,191,216,168)(89,192,217,155)(90,193,218,156)(91,194,219,157)(92,195,220,158)(93,196,221,159)(94,183,222,160)(95,184,223,161)(96,185,224,162)(97,186,211,163)(98,187,212,164), (1,63,8,70)(2,64,9,57)(3,65,10,58)(4,66,11,59)(5,67,12,60)(6,68,13,61)(7,69,14,62)(15,161,22,168)(16,162,23,155)(17,163,24,156)(18,164,25,157)(19,165,26,158)(20,166,27,159)(21,167,28,160)(29,222,36,215)(30,223,37,216)(31,224,38,217)(32,211,39,218)(33,212,40,219)(34,213,41,220)(35,214,42,221)(43,115,50,122)(44,116,51,123)(45,117,52,124)(46,118,53,125)(47,119,54,126)(48,120,55,113)(49,121,56,114)(71,209,78,202)(72,210,79,203)(73,197,80,204)(74,198,81,205)(75,199,82,206)(76,200,83,207)(77,201,84,208)(85,153,92,146)(86,154,93,147)(87,141,94,148)(88,142,95,149)(89,143,96,150)(90,144,97,151)(91,145,98,152)(99,140,106,133)(100,127,107,134)(101,128,108,135)(102,129,109,136)(103,130,110,137)(104,131,111,138)(105,132,112,139)(169,189,176,196)(170,190,177,183)(171,191,178,184)(172,192,179,185)(173,193,180,186)(174,194,181,187)(175,195,182,188)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,23,113,179)(2,22,114,178)(3,21,115,177)(4,20,116,176)(5,19,117,175)(6,18,118,174)(7,17,119,173)(8,16,120,172)(9,15,121,171)(10,28,122,170)(11,27,123,169)(12,26,124,182)(13,25,125,181)(14,24,126,180)(29,106,148,201)(30,105,149,200)(31,104,150,199)(32,103,151,198)(33,102,152,197)(34,101,153,210)(35,100,154,209)(36,99,141,208)(37,112,142,207)(38,111,143,206)(39,110,144,205)(40,109,145,204)(41,108,146,203)(42,107,147,202)(43,183,58,167)(44,196,59,166)(45,195,60,165)(46,194,61,164)(47,193,62,163)(48,192,63,162)(49,191,64,161)(50,190,65,160)(51,189,66,159)(52,188,67,158)(53,187,68,157)(54,186,69,156)(55,185,70,155)(56,184,57,168)(71,214,134,93)(72,213,135,92)(73,212,136,91)(74,211,137,90)(75,224,138,89)(76,223,139,88)(77,222,140,87)(78,221,127,86)(79,220,128,85)(80,219,129,98)(81,218,130,97)(82,217,131,96)(83,216,132,95)(84,215,133,94), (1,192)(2,193)(3,194)(4,195)(5,196)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,43)(26,44)(27,45)(28,46)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,71)(42,72)(57,180)(58,181)(59,182)(60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,176)(68,177)(69,178)(70,179)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(97,105)(98,106)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,153)(128,154)(129,141)(130,142)(131,143)(132,144)(133,145)(134,146)(135,147)(136,148)(137,149)(138,150)(139,151)(140,152)(197,215)(198,216)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222)(205,223)(206,224)(207,211)(208,212)(209,213)(210,214), (1,199,120,111)(2,200,121,112)(3,201,122,99)(4,202,123,100)(5,203,124,101)(6,204,125,102)(7,205,126,103)(8,206,113,104)(9,207,114,105)(10,208,115,106)(11,209,116,107)(12,210,117,108)(13,197,118,109)(14,198,119,110)(15,37,178,149)(16,38,179,150)(17,39,180,151)(18,40,181,152)(19,41,182,153)(20,42,169,154)(21,29,170,141)(22,30,171,142)(23,31,172,143)(24,32,173,144)(25,33,174,145)(26,34,175,146)(27,35,176,147)(28,36,177,148)(43,77,65,133)(44,78,66,134)(45,79,67,135)(46,80,68,136)(47,81,69,137)(48,82,70,138)(49,83,57,139)(50,84,58,140)(51,71,59,127)(52,72,60,128)(53,73,61,129)(54,74,62,130)(55,75,63,131)(56,76,64,132)(85,188,213,165)(86,189,214,166)(87,190,215,167)(88,191,216,168)(89,192,217,155)(90,193,218,156)(91,194,219,157)(92,195,220,158)(93,196,221,159)(94,183,222,160)(95,184,223,161)(96,185,224,162)(97,186,211,163)(98,187,212,164), (1,63,8,70)(2,64,9,57)(3,65,10,58)(4,66,11,59)(5,67,12,60)(6,68,13,61)(7,69,14,62)(15,161,22,168)(16,162,23,155)(17,163,24,156)(18,164,25,157)(19,165,26,158)(20,166,27,159)(21,167,28,160)(29,222,36,215)(30,223,37,216)(31,224,38,217)(32,211,39,218)(33,212,40,219)(34,213,41,220)(35,214,42,221)(43,115,50,122)(44,116,51,123)(45,117,52,124)(46,118,53,125)(47,119,54,126)(48,120,55,113)(49,121,56,114)(71,209,78,202)(72,210,79,203)(73,197,80,204)(74,198,81,205)(75,199,82,206)(76,200,83,207)(77,201,84,208)(85,153,92,146)(86,154,93,147)(87,141,94,148)(88,142,95,149)(89,143,96,150)(90,144,97,151)(91,145,98,152)(99,140,106,133)(100,127,107,134)(101,128,108,135)(102,129,109,136)(103,130,110,137)(104,131,111,138)(105,132,112,139)(169,189,176,196)(170,190,177,183)(171,191,178,184)(172,192,179,185)(173,193,180,186)(174,194,181,187)(175,195,182,188) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,23,113,179),(2,22,114,178),(3,21,115,177),(4,20,116,176),(5,19,117,175),(6,18,118,174),(7,17,119,173),(8,16,120,172),(9,15,121,171),(10,28,122,170),(11,27,123,169),(12,26,124,182),(13,25,125,181),(14,24,126,180),(29,106,148,201),(30,105,149,200),(31,104,150,199),(32,103,151,198),(33,102,152,197),(34,101,153,210),(35,100,154,209),(36,99,141,208),(37,112,142,207),(38,111,143,206),(39,110,144,205),(40,109,145,204),(41,108,146,203),(42,107,147,202),(43,183,58,167),(44,196,59,166),(45,195,60,165),(46,194,61,164),(47,193,62,163),(48,192,63,162),(49,191,64,161),(50,190,65,160),(51,189,66,159),(52,188,67,158),(53,187,68,157),(54,186,69,156),(55,185,70,155),(56,184,57,168),(71,214,134,93),(72,213,135,92),(73,212,136,91),(74,211,137,90),(75,224,138,89),(76,223,139,88),(77,222,140,87),(78,221,127,86),(79,220,128,85),(80,219,129,98),(81,218,130,97),(82,217,131,96),(83,216,132,95),(84,215,133,94)], [(1,192),(2,193),(3,194),(4,195),(5,196),(6,183),(7,184),(8,185),(9,186),(10,187),(11,188),(12,189),(13,190),(14,191),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,43),(26,44),(27,45),(28,46),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,71),(42,72),(57,180),(58,181),(59,182),(60,169),(61,170),(62,171),(63,172),(64,173),(65,174),(66,175),(67,176),(68,177),(69,178),(70,179),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,99),(92,100),(93,101),(94,102),(95,103),(96,104),(97,105),(98,106),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,155),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(127,153),(128,154),(129,141),(130,142),(131,143),(132,144),(133,145),(134,146),(135,147),(136,148),(137,149),(138,150),(139,151),(140,152),(197,215),(198,216),(199,217),(200,218),(201,219),(202,220),(203,221),(204,222),(205,223),(206,224),(207,211),(208,212),(209,213),(210,214)], [(1,199,120,111),(2,200,121,112),(3,201,122,99),(4,202,123,100),(5,203,124,101),(6,204,125,102),(7,205,126,103),(8,206,113,104),(9,207,114,105),(10,208,115,106),(11,209,116,107),(12,210,117,108),(13,197,118,109),(14,198,119,110),(15,37,178,149),(16,38,179,150),(17,39,180,151),(18,40,181,152),(19,41,182,153),(20,42,169,154),(21,29,170,141),(22,30,171,142),(23,31,172,143),(24,32,173,144),(25,33,174,145),(26,34,175,146),(27,35,176,147),(28,36,177,148),(43,77,65,133),(44,78,66,134),(45,79,67,135),(46,80,68,136),(47,81,69,137),(48,82,70,138),(49,83,57,139),(50,84,58,140),(51,71,59,127),(52,72,60,128),(53,73,61,129),(54,74,62,130),(55,75,63,131),(56,76,64,132),(85,188,213,165),(86,189,214,166),(87,190,215,167),(88,191,216,168),(89,192,217,155),(90,193,218,156),(91,194,219,157),(92,195,220,158),(93,196,221,159),(94,183,222,160),(95,184,223,161),(96,185,224,162),(97,186,211,163),(98,187,212,164)], [(1,63,8,70),(2,64,9,57),(3,65,10,58),(4,66,11,59),(5,67,12,60),(6,68,13,61),(7,69,14,62),(15,161,22,168),(16,162,23,155),(17,163,24,156),(18,164,25,157),(19,165,26,158),(20,166,27,159),(21,167,28,160),(29,222,36,215),(30,223,37,216),(31,224,38,217),(32,211,39,218),(33,212,40,219),(34,213,41,220),(35,214,42,221),(43,115,50,122),(44,116,51,123),(45,117,52,124),(46,118,53,125),(47,119,54,126),(48,120,55,113),(49,121,56,114),(71,209,78,202),(72,210,79,203),(73,197,80,204),(74,198,81,205),(75,199,82,206),(76,200,83,207),(77,201,84,208),(85,153,92,146),(86,154,93,147),(87,141,94,148),(88,142,95,149),(89,143,96,150),(90,144,97,151),(91,145,98,152),(99,140,106,133),(100,127,107,134),(101,128,108,135),(102,129,109,136),(103,130,110,137),(104,131,111,138),(105,132,112,139),(169,189,176,196),(170,190,177,183),(171,191,178,184),(172,192,179,185),(173,193,180,186),(174,194,181,187),(175,195,182,188)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 11 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 13 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 14 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 27 | 27 | 0 | 0 | 0 | 0 |
| 17 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,11,0,0,0,0,21,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[2,13,0,0,0,0,2,27,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[1,27,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[27,17,0,0,0,0,27,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,17] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊2D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.432+ (1+4) | C23.D14 | Dic7⋊4D4 | C4.Dic14 | D7×C4⋊C4 | C23.21D14 | C4×C7⋊D4 | D4×Dic7 | C23.18D14 | C28⋊2D4 | C7×C4⋊D4 | C4⋊D4 | C28 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 3 | 4 | 4 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{43}2_+^{(1+4)} % in TeX
G:=Group("C14.43ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1067);
// by ID
G=gap.SmallGroup(448,1067);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,100,1571,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*b^2,e^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,e*c*e^-1=a^7*c,e*d*e^-1=b^2*d>;
// generators/relations
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